On optional stopping of some exponential martingales for Lévy processes with or without reflection

Sören Asmussen and Offer Kella

Abstract:: Kella Whitt (1992) introduced a martingale $\{M_t\}$ for processes of the form $Z_t=X_t+Y_t$ where $\{X_t\}$ is a L\'evy process and $Y_t$ satisfies certain; regularity conditions. In particular this provides a martingale for the case where $Y_t=L_t$ where $L_t$ is the local time at zero of the corresponding reflected L\'evy process. In this case $\{M_t\}$ involves, among others, the L\'evy exponent $\varphi(\alpha)$ and $L_t$. In this paper, conditions for optional stopping of $\{M_t\}$ at $\tau$ are given. The conditions depend on the signs of $\alpha$ and $\varphi(\alpha)$. In some cases optional stopping is always permissible. In others,; the conditions involve the well known necessary and sufficient condition for optional stopping of the Wald martingale $\{e^{\alpha X_t-t\phi(\alpha)}\}$, namely that $\tilde{P}(\tau<\infty) $ $=1$ where $\tilde{P}$ corresponds to a suitable exponentially tilted L\'evy process.
: Key words:; exponential change of measure, Lévy process, local time, stopping time, Wald martingale